The First Order Necessary Condition Of Optimal Control Problem For A Class Of P-laplace Equation

Optimal control of control constrained partial differential equations is a veryrich subject from the theoretical and applied points of view. It is the mathematicalmodel of fluid control, image processing, biological population control, spacecraftdesign, mechanical design and shape optimization and many other fields.Based on the introduction of the background and the research status of PDEconstrained optimal control problems, we consider the optimal control problem ofa p-Laplace partial differential equation with a Dirichlet boundary after a lot ofbasic knowledge is given. After restricting that the control variable is nonnegative,the main work of this paper is as follow:(1) We demonstrate the existence of the weak solution of the p-Laplaceequation with the Dirichlet boundary condition by the variational method, andprovide the equations which are met by the directional derivative of the controlvariable. Then we also prove the existence of the strong solution of the nonlinearp-Laplace equation. Besides the above results, with the similar method, weintroduce a kind of the differential inclusion problems of the p-Laplace equationand prove the existence of the strong solution of it.(2) We define the corresponding optimal control problem. It is said to be thatwe define a objective function, with the purpose of describing the set of thevariables which make the objective function equivalent to its minimum. Then weprove the existence of the solution of the optimal control problem.(3) This paper provides the first order necessary condition of the controlproblem of the p-Laplace equation and discusses some special cases.